nLab Sandbox

Idea

In algebra one speaks of internal direct sums for direct sums of subobjects of a given object that are themselves again canonically subobjects of that object.

In contrast to this situation, the ordinary direct sum is sometimes called the external direct sum. $\mathcal{n}$

Definition

For modules

Traditionally this is considered for modules $N$ over a ring $R$ and stated as follows (see the references below):

Let $\big\{N_i \subset N\big\}_{i \in I}$ be an indexed set of of submodules $N_i \hookrightarrow N$. Then a submodule

(1)$$\oplus^{int}_{i \in I} N_i \,\xhookrightarrow{\;\; q \;\;}\, N$$

is called their internal direct sum if the following condition holds

• For every $n \in \bigoplus^{int}_{i \in I} N_i$ there is a unique $I$-tuple $(n_i \in N_i)_{i \in I}$ such that their sum in $N$ is $\sum_i n_i \,=\, n$.

Of course, this is equivalent to the condition:

• $\bigoplus^{int}_{i \in I} N_i \,\simeq\, \bigoplus_{i \in I} N_i$ is isomorphism to the abstract (external) direct sum and $q$ in (1) is the universal morphism induced from the inclusions $N_i \hookrightarrow N$.

In this form the definition clearly generalizes:

Generally

Given an object $B$ and a family of subobjects $A_i$ of $B$ (or more generally a family of morphisms $A_i \to B$, or equivalently a map $\coprod_i A_i \to B$), suppose that the direct sum $\bigoplus_i A_i$ exists. Suppose further that the map $\coprod_i A_i \to B$ factors through the map $\coprod_i A_i \to \bigoplus_i A_i$ (which means that it factors uniquely if $\coprod_i A_i \to \bigoplus_i A_i$ is epic, as it must be in a regular category). Finally, suppose that the (or a) quotient map $\bigoplus_i A_i \to B$ is an isomorphism. Then we say that $B$ is the internal direct sum of the $A_i$.

In contrast, the abstractly defined direct sum $\bigoplus_i A_i$ may be called an external direct sum. These terms are usually used with concrete categories where the $A_i$ may either be given independently (for an external direct sum) or as subsets of some ambient space (either $B$ or something of which $B$ is a subset) for an internal direct sum. In too abstract a context, there is no difference: on the one hand, any internal direct sum is a fortiori isomorphic to any external direct sum; on the other hand, given an external direct sum, there is a natural map $\coprod_i A_i \to \bigoplus_i A_i$, relative to which the external direct sum is an internal direct sum.

References

References for the tradition definition

• ProofWiki, Internal Direct Sum of Modules

• Lia-Ming Liou, External direct sum and Internal direct sum of vector spaces, lecture notes [pdf]